Toniolo and Linder, Equation (10+)

Average Error: 32.0 → 13.1

Time: 27.8s

Precision: 64

Math

\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.14299624566779908 \cdot 10^{-185} \lor \neg \left(t \le 1.182101435787054 \cdot 10^{-185}\right):\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array}\]

C

double f(double t, double l, double k) {
        double r134131 = 2.0;
        double r134132 = t;
        double r134133 = 3.0;
        double r134134 = pow(r134132, r134133);
        double r134135 = l;
        double r134136 = r134135 * r134135;
        double r134137 = r134134 / r134136;
        double r134138 = k;
        double r134139 = sin(r134138);
        double r134140 = r134137 * r134139;
        double r134141 = tan(r134138);
        double r134142 = r134140 * r134141;
        double r134143 = 1.0;
        double r134144 = r134138 / r134132;
        double r134145 = pow(r134144, r134131);
        double r134146 = r134143 + r134145;
        double r134147 = r134146 + r134143;
        double r134148 = r134142 * r134147;
        double r134149 = r134131 / r134148;
        return r134149;
}

↓

double f(double t, double l, double k) {
        double r134150 = t;
        double r134151 = -5.142996245667799e-185;
        bool r134152 = r134150 <= r134151;
        double r134153 = 1.1821014357870538e-185;
        bool r134154 = r134150 <= r134153;
        double r134155 = !r134154;
        bool r134156 = r134152 || r134155;
        double r134157 = 2.0;
        double r134158 = cbrt(r134150);
        double r134159 = r134158 * r134158;
        double r134160 = 3.0;
        double r134161 = 2.0;
        double r134162 = r134160 / r134161;
        double r134163 = pow(r134159, r134162);
        double r134164 = l;
        double r134165 = cbrt(r134164);
        double r134166 = r134165 * r134165;
        double r134167 = r134163 / r134166;
        double r134168 = r134163 / r134165;
        double r134169 = pow(r134158, r134160);
        double r134170 = r134169 / r134164;
        double r134171 = k;
        double r134172 = sin(r134171);
        double r134173 = r134170 * r134172;
        double r134174 = r134168 * r134173;
        double r134175 = tan(r134171);
        double r134176 = r134174 * r134175;
        double r134177 = 1.0;
        double r134178 = r134171 / r134150;
        double r134179 = pow(r134178, r134157);
        double r134180 = r134177 + r134179;
        double r134181 = r134180 + r134177;
        double r134182 = r134176 * r134181;
        double r134183 = r134167 * r134182;
        double r134184 = r134157 / r134183;
        double r134185 = 1.0;
        double r134186 = -1.0;
        double r134187 = pow(r134186, r134160);
        double r134188 = r134185 / r134187;
        double r134189 = pow(r134188, r134177);
        double r134190 = cbrt(r134186);
        double r134191 = 9.0;
        double r134192 = pow(r134190, r134191);
        double r134193 = 3.0;
        double r134194 = pow(r134150, r134193);
        double r134195 = pow(r134172, r134161);
        double r134196 = r134194 * r134195;
        double r134197 = r134192 * r134196;
        double r134198 = cos(r134171);
        double r134199 = pow(r134164, r134161);
        double r134200 = r134198 * r134199;
        double r134201 = r134197 / r134200;
        double r134202 = r134189 * r134201;
        double r134203 = r134157 * r134202;
        double r134204 = pow(r134171, r134161);
        double r134205 = r134204 * r134150;
        double r134206 = r134195 * r134205;
        double r134207 = r134192 * r134206;
        double r134208 = r134207 / r134200;
        double r134209 = r134189 * r134208;
        double r134210 = r134203 + r134209;
        double r134211 = r134157 / r134210;
        double r134212 = r134156 ? r134184 : r134211;
        return r134212;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if t < -5.142996245667799e-185 or 1.1821014357870538e-185 < t

   1. Initial program 27.2

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 27.4

      \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied unpow-prod-down 27.4

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   5. Applied times-frac 19.6

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   6. Applied associate-*l* 17.6

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   7. Using strategy rm

   8. Applied add-cube-cbrt 17.6

      \[\leadsto \frac{2}{\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   9. Applied sqr-pow 17.6

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   10. Applied times-frac 13.0

       \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}}\right)} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   11. Applied associate-*l* 12.1

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   12. Using strategy rm

   13. Applied associate-*l* 10.4

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   14. Using strategy rm

   15. Applied associate-*l* 9.0

       \[\leadsto \frac{2}{\color{blue}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}}\]

   if -5.142996245667799e-185 < t < 1.1821014357870538e-185

   1. Initial program 64.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 64.0

      \[\leadsto \frac{2}{\left(\left(\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   4. Applied unpow-prod-down 64.0

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   5. Applied times-frac 64.0

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   6. Applied associate-*l* 64.0

      \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

   7. Taylor expanded around -inf 40.5

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.14299624566779908 \cdot 10^{-185} \lor \neg \left(t \le 1.182101435787054 \cdot 10^{-185}\right):\\ \;\;\;\;\frac{2}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \left(\left(\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{\left(\frac{3}{2}\right)}}{\sqrt[3]{\ell}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \left({\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({t}^{3} \cdot {\left(\sin k\right)}^{2}\right)}{\cos k \cdot {\ell}^{2}}\right) + {\left(\frac{1}{{-1}^{3}}\right)}^{1} \cdot \frac{{\left(\sqrt[3]{-1}\right)}^{9} \cdot \left({\left(\sin k\right)}^{2} \cdot \left({k}^{2} \cdot t\right)\right)}{\cos k \cdot {\ell}^{2}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020027 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))